Remarks on replica diagonal collective field condensations in SYK

Abstract

In the Sachdev-Ye-Kitaev model with generic order $q \ge 4$ random couplings, we compute the critical temperature relating the Majorana fermions high temperature perturbative vacuum to the vacuum where the replica diagonal collective field $G(τ, τ')$ condenses. We study, by a finite temperature diagrammatic analysis, the effective action of an auxiliary Hubbard-Stratonovich bilocal field related to $G(τ, τ')$ in the large $N$ limit. Subtelties that arise in switching from the operatorial to the functional integral representation of the SYK thermal partition function are also discussed.

The perturbative expansion has not kept into account the necessary subtractions